Pvs stands for parachute vortex shedding. The pvs is the cause of resonance of the system. At high altitudes, the free-stream density is sufficiently low to result in a relatively small ratio of apparent mass to payload mass, causing the system center of gravity to be located very close to the payload. This result in a long momentum arm between the canopy and the system center of gravity, such that the pvs can dominate the dynamics of the complete system.

$f_{pvs}= \frac{S_t . V_{term}}{d_{ratio} . d_o}$

$S_t$ stands for Strouhal number and $d_o$ is the nominal diameter of the parachute.

• In our modelisation we won't consider the PVS oscillation due to the mass of the rover 2Kg and the altitude.
• The parachute that we'll implement will reduce the terminal velocity in 90%.
• For an altitude smaller than 1,6 Km we can apply the general dragging formula $2.W=c.\rho.V_{term}^2.A$ where W stands for the weight, $\rho$ is the air density, c the dragging coefficient, $V_{term}$ the terminal velocity and A the area of the parachute.
• The recomending $V_{term}$ or landing velocity is between $3.5{m}{s}$ to $4.5{m}{s}$.
• The dragging coefficient for a half-sphere $c$ is 0.47.
• The density of the air increase with low temperatures; at -10°C $\rho=1.34 {Kg}{m^3}$
• The area of the parachute can be obtained: $A=\frac{2.g.M_{rover}}{c.\rho.V_{term}^2}$
• For a round parachute the diameter is obtained: $d=\sqrt{\frac{4.A}{\pi}}$
• The spill hole to gain stability should have a diameter of 20% of the total diameter of the canopy.
• If the area and/or the diameter of the canopy are fixed, we can change the diameter of the spill hole in function of the terminal velocity, reducing the total area in the formula above.
• Dynamics of free fall with air resistance, says that a body will arrive to the terminal velocity after 5 seconds, with out depending on the altitude.

References:

• F. M. WHITE and D. F. WOLF. “A theory of three-dimensional parachute dynamic stability.” Journal of Aircraft, Vol. 5, No. 1 (1968), pp. 86-92. doi: 10.2514/3.43912
• A Method to Characterize Parachute System Instability Modes by Michael Bernatovich Dr. Ian Clark 5/6/2013